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A recurrence theorem for square-integrable martingales

Gerold Alsmeyer (1994)

Studia Mathematica

Let ( M n ) n 0 be a zero-mean martingale with canonical filtration ( n ) n 0 and stochastically L 2 -bounded increments Y 1 , Y 2 , . . . , which means that P ( | Y n | > t | n - 1 ) 1 - H ( t ) a.s. for all n ≥ 1, t > 0 and some square-integrable distribution H on [0,∞). Let V 2 = n 1 E ( Y n 2 | n - 1 ) . It is the main result of this paper that each such martingale is a.s. convergent on V < ∞ and recurrent on V = ∞, i.e. P ( M n [ - c , c ] i . o . | V = ) = 1 for some c > 0. This generalizes a recent result by Durrett, Kesten and Lawler [4] who consider the case of only finitely many square-integrable increment distributions....

Application des lois non paramétriques dans les systèmes d'attente et la théorie de renouvellement

Smail Adjabi, Karima Lagha, Amar Aïssani (2010)

RAIRO - Operations Research

Les distributions non paramétriques de survie trouvent, de plus en plus, des applications dans des domaines très variés, à savoir: théorie de fiabilité et analyse de survie, files d'attente, maintenance, gestion de stock, théorie de l'économie, ... L'objet de ce travail est d'utiliser les bornes inférieures et supérieures (en terme de la moyenne) des fonctions de fiabilité appartenant aux classes de distribution de type IFR, DFR, NBU et NWU, présentées par Sengupta (1994), pour l'évaluation de...

Application des lois non paramétriques dans les systèmes d’attente et la théorie de renouvellement

Smail Adjabi, Karima Lagha, Amar Aïssani (2004)

RAIRO - Operations Research - Recherche Opérationnelle

Les distributions non paramétriques de survie trouvent, de plus en plus, des applications dans des domaines très variés, à savoir : théorie de fiabilité et analyse de survie, files d’attente, maintenance, gestion de stock, théorie de l’économie, ... L’objet de ce travail est d’utiliser les bornes inférieures et supérieures (en terme de la moyenne) des fonctions de fiabilité appartenant aux classes de distribution de type I F R , D F R , N B U et N W U , présentées par Sengupta (1994), pour l’évaluation de certaines caractéristiques....

Approximation of bivariate Markov chains by one-dimensional diffusion processes

Daniela Kuklíková (1978)

Aplikace matematiky

The paper deals with several questions of the diffusion approximation. The goal of this paper is to create the general method of reducting the dimension of the model with the aid of the diffusion approximation. Especially, two dimensional random variables are approximated by one-dimensional diffusion process by replacing one of its coordinates by a certain characteristic, e.g. by its stationary expectation. The suggested method is used for several different systems. For instance, the method is applicable...

Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes

Bernard Roynette, Pierre Vallois, Agnès Volpi (2007)

ESAIM: Probability and Statistics

Let (Xt, t ≥ 0) be a Lévy process started at 0, with Lévy measure ν. We consider the first passage time Tx of (Xt, t ≥ 0) to level x > 0, and Kx := XTx - x the overshoot and Lx := x- XTx- the undershoot. We first prove that the Laplace transform of the random triple (Tx,Kx,Lx) satisfies some kind of integral equation. Second, assuming that ν admits exponential moments, we show that ( T x ˜ , K x , L x ) converges in distribution as x → ∞, where T x ˜ denotes a suitable renormalization of Tx.


Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes

Bernard Roynette, Pierre Vallois, Agnès Volpi (2008)

ESAIM: Probability and Statistics

Let ( X t , t 0 ) be a Lévy process started at 0 , with Lévy measure ν . We consider the first passage time T x of ( X t , t 0 ) to level x &gt; 0 , and K x : = X T x - 𝑥 the overshoot and L x : = x - X T 𝑥 - the undershoot. We first prove that the Laplace transform of the random triple ( T x , K x , L x ) satisfies some kind of integral equation. Second, assuming that ν admits exponential moments, we show that ( T x ˜ , K x , L x ) converges in distribution as x , where T x ˜ denotes a suitable renormalization of T x .

Binomial ARMA count series from renewal processes

Sergiy Koshkin, Yunwei Cui (2012)

Discussiones Mathematicae Probability and Statistics

This paper describes a new method for generating stationary integer-valued time series from renewal processes. We prove that if the lifetime distribution of renewal processes is nonlattice and the probability generating function is rational, then the generated time series satisfy causal and invertible ARMA type stochastic difference equations. The result provides an easy method for generating integer-valued time series with ARMA type autocovariance functions. Examples of generating binomial ARMA(p,p-1)...

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